Open Access

Second Moment Convergence Rates for Uniform Empirical Processes

Journal of Inequalities and Applications20102010:972324

Received: 21 May 2010

Accepted: 19 August 2010

Published: 23 August 2010


Let be a sequence of independent and identically distributed -distributed random variables. Define the uniform empirical process as , . In this paper, we get the exact convergence rates of weighted infinite series of .

1. Introduction and Main Results

Let be a sequence of independent and identically distributed (i.i.d.) random variables with zero mean. Set for , and . Hsu and Robbins [1] introduced the concept of complete convergence. They showed that

if and . The converse part was proved by the study of Erdös in [2]. Obviously, the sum in (1.1) tends to infinity as . Many authors studied the exact rates in terms of (cf. [35]). Chow [6] studied the complete convergence of , . Recently, Liu and Lin [7] introduced a new kind of complete moment convergence which is interesting, and got the precise rate of it as follows.

Theorem A.

Suppose that is a sequence of i.i.d. random variables, then

holds, if and only if , , and .

Other than partial sums, many authors investigated precise rates in some different cases, such as U-statistics (cf. [8, 9]) and self-normalized sums (cf. [10, 11]). Zhang and Yang [12] extended the precise asymptotic results to the uniform empirical process. We suppose is the sample of random variables and is the empirical distribution function of it. Denote the uniform empirical process by , , and the norm of a function on by . Let , be the Brownian bridge. We present one result of Zhang and Yang [12] as follows.

Theorem B.

For any , one has

Inspired by the above conclusions, we consider second moment convergence rates for the uniform empirical process in the law of iterated logarithm and the law of the logarithm. Throughout this paper, let denote a positive constant whose values can be different from one place to another. will denote the largest integer . The following two theorems are our main results.

Theorem 1.1.

For , one has

Theorem 1.2.

For , one has

Remark 1.3.

It is well known that , (see Csörgő and Révész [13, page 43]). Therefore, by Fubini's theorem we have

Consequently, explicit results of (1.4) and (1.5) can be calculated further.

2. The Proofs

In order to prove Theorem 1.1, we present several propositions first.

Proposition 2.1.

For , , one has


We calculate that

Proposition 2.2.

For , one has


Following [4], set , where . Write
It is wellknown that (see Csörgő and Révész [13, page 17]). By continuous mapping theorem, we have . As a result, it follows that
Using the Toeplitz's lemma (see Stout [14, pages 120-121]), we can get . For , it is obvious that
Notice that , for a small . Via the similar argument in [4] we have
From Kiefer and Wolfowitz [15], we have

From (2.6), (2.7), and (2.9), we get . Proposition 2.2 has been proved.

Proposition 2.3.

For , one has


The calculation here is analogous to (2.1), so it is omitted here.

Proposition 2.4.

For , one has


Like [4] and Proposition 2.2, we divide the summation into two parts,
First, consider ,
Since means , it follows
By Lemma in Zhang and Yang [12], we have . For , it is easy to get
In the same way, by the inequality , we can get

Put the three parts together, we get that uniformly in as . Using Toeplitz's lemma again, we have

In the sequel, we verify . It is easy to see that
We estimate first, by noticing and (2.8), it follows

Therefore, we get . So far, we only need to prove . Use the inequality again and follow the proof of , we can get this result. The proof of the proposition is completed now.

Proof of Theorem 1.1.

According to Fubini's theorem, it is easy to get
for . Therefore, we have
From Proposition 2.1– 2.4, we have

Proof of Theorem 1.2.

From (2.19), we have
Via the similar argument in Proposition 2.1 and 2.2,
Also, by the analogous proof of Proposition 2.3 and 2.4,

Combine (2.22), (2.23), and (2.24)together, we get the result of Theorem 1.2.



This work was supported by NSFC (No. 10771192) and ZJNSF (No. J20091364).

Authors’ Affiliations

Department of Mathematics, Zhejiang University, Hangzhou, China


  1. Hsu PL, Robbins H: Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America 1947, 33: 25–31. 10.1073/pnas.33.2.25MathSciNetView ArticleMATHGoogle Scholar
  2. Erdös P: On a theorem of Hsu and Robbins. Annals of Mathematical Statistics 1949, 20: 286–291. 10.1214/aoms/1177730037MathSciNetView ArticleMATHGoogle Scholar
  3. Chen R: A remark on the tail probability of a distribution. Journal of Multivariate Analysis 1978, 8(2):328–333. 10.1016/0047-259X(78)90084-2MathSciNetView ArticleMATHGoogle Scholar
  4. Gut A, Spătaru A: Precise asymptotics in the Baum-Katz and Davis laws of large numbers. Journal of Mathematical Analysis and Applications 2000, 248(1):233–246. 10.1006/jmaa.2000.6892MathSciNetView ArticleMATHGoogle Scholar
  5. Heyde CC: A supplement to the strong law of large numbers. Journal of Applied Probability 1975, 12: 173–175. 10.2307/3212424MathSciNetView ArticleMATHGoogle Scholar
  6. Chow YS: On the rate of moment convergence of sample sums and extremes. Bulletin of the Institute of Mathematics 1988, 16(3):177–201.MathSciNetMATHGoogle Scholar
  7. Liu W, Lin Z: Precise asymptotics for a new kind of complete moment convergence. Statistics & Probability Letters 2006, 76(16):1787–1799. 10.1016/j.spl.2006.04.027MathSciNetView ArticleMATHGoogle Scholar
  8. Fu K-A: Asymptotics for the moment convergence of -statistics in LIL. Journal of Inequalities and Applications 2010, 2010:-8.MathSciNetMATHGoogle Scholar
  9. Yan JG, Su C: Precise asymptotics of -statistics. Acta Mathematica Sinica 2007, 50(3):517–526.MathSciNetMATHGoogle Scholar
  10. Pang T-X, Zhang L-X, Wang JF: Precise asymptotics in the self-normalized law of the iterated logarithm. Journal of Mathematical Analysis and Applications 2008, 340(2):1249–1262. 10.1016/j.jmaa.2007.09.054MathSciNetView ArticleMATHGoogle Scholar
  11. Zang Q-P: A limit theorem for the moment of self-normalized sums. Journal of Inequalities and Applications 2009, 2009:-10.MathSciNetView ArticleMATHGoogle Scholar
  12. Zhang Y, Yang X-Y: Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process. Statistics & Probability Letters 2008, 78(9):1051–1055. 10.1016/j.spl.2007.09.063MathSciNetView ArticleMATHGoogle Scholar
  13. Csörgő M, Révész P: Strong Approximations in Probability and Statistics, Probability and Mathematical Statistics. Academic Press, New York, NY, USA; 1981:284.MATHGoogle Scholar
  14. Stout WF: Almost Sure Convergence. Academic Press, New York, NY, USA; 1974:x+381. Probability and Mathematical Statistics, Vol. 2MATHGoogle Scholar
  15. Kiefer J, Wolfowitz J: On the deviations of the empiric distribution function of vector chance variables. Transactions of the American Mathematical Society 1958, 87: 173–186. 10.1090/S0002-9947-1958-0099075-1MathSciNetView ArticleMATHGoogle Scholar


© Y.-Y. Chen and L.-X. Zhang. 2010

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